Subtracting Multi-Digit Numbers

This lesson is where the child subtracts multi-digit whole numbers fluently, including across zeros. Each place is ten times the one to its right, and that single rule generates the whole system all the way to a million and, later, all the way down past the decimal point.

Watch for large numbers read digit by digit. "Four hundred and seven thousand" is a structure; "four zero seven" is a phone number.

Have them say the number aloud before working with it. Most errors with big numbers are reading errors that the arithmetic then faithfully preserves.

The slip this lesson targets is taking the smaller digit from the bigger one whichever row it sits in: Nia writes 7 − 1 = 6 in the ones of 6,341 − 2,517. Ask which digit is on top before any column is subtracted. The second slip is Dev's: a column that lends a unit away is one smaller afterwards, so 6,000 − 2,346 finishes 5 − 2 in the thousands, not 6 − 2. Both produce an answer that is a whole thousand out, which is exactly what adding the answer back catches.

Unbundling, several columns at once — When a column has too little to subtract from, open one unit from the column to its left: a thousand becomes ten hundreds, a hundred becomes ten tens. The value of the number is unchanged — only its packaging. This is the whole of multi-digit subtraction.

Unbundling across zeros — In 5,003 - 2,478 the tens and hundreds are both empty, so you must go left to the thousands and cascade: 5,003 becomes 4 thousands, 9 hundreds, 9 tens and 13 ones. Do the whole cascade first, writing every change down, and only then start subtracting.

Check by adding back — If 5,003 - 2,478 = 2,525, then 2,525 + 2,478 should return 5,003. It does. This check uses a different operation, so it is genuinely independent of the work you just did — unlike re-reading your own subtraction, which tends to repeat the same mistake.

When counting up is easier — For 5,003 - 4,987 the numbers are close, so counting up beats unbundling: 4,987 to 5,000 is 13, then 3 more is 16. Recognizing when the algorithm is the wrong tool is part of being genuinely fluent rather than merely trained.

Break open the next unit along — 50,000 − 23,456. Ones: 0 − 6. Nothing to break next door, so keep going left until you find something. Break the 5 ten-thousands into 4 ten-thousands and 10 thousands, then cascade down. Now every column has enough. 26,544. Check it: 26,544 + 23,456 = 50,000.